The present invention relates to signals such as radar signals that are used for range and direction finding and for imaging surfaces and objects and, more particularly, to a class of such signals whose length and bandwidth are parametrized independently. These signals may be used by range and direction finding devices, such as radars and their acoustic equivalents, that have single or multiple transmitters and receivers.
Pulsed signals such as radio-frequency (RF) signals and acoustic signals commonly are used for determining the distance and direction to a target. Reflections of the signals from the target are received, and the distance to the target is inferred as the product of the round-trip travel time and the speed of the signals in the medium in which the signals propagate to and from the target. The direction of arrival (DOA) is identified e.g. from differential arrival times at the receivers of a receiver array. Co-pending U.S. patent application Ser. No. 10/571,693, that is incorporated by reference for all purposes as if fully set forth herein, teaches a method of transmitting acoustic signals from the top of a silo and receiving echoes of the signals from the contents of the silo at the top of the silo to measure the shape of the top of the contents of the silo and so infer the volume of the contents of the silo. FIG. 1 shows one such silo 100, in cross-section. This application of pulsed signal range and direction finding is complicated by challenges including:
(i) The upper surface 104a or 104b of the silo contents 102 can be at a variable distance from the acoustic transmitters and receivers 106. This distance varies in a wide range: from tens of centimeters (104a) to tens of meters (104b). This creates the following sub-challenges:
i-1: a need for signal-to-noise-ratio (SNR) enhancement of weak arrived reflected pulses.
i-2: a need for pulses of different lengths
The need for SNR enhancement follows, in addition to the mentioned wide dynamic range of the acoustic paths, from the following issues:                (A) Specific silo contents surface geometry and/or specific geometry of silo walls and/or due to kinds of granularity of the silo contents. In particular, if due to specific environmental factors named above the geometrically reflected acoustic ray trajectories miss the receivers and only weak diffuse scattered rays approach the receivers, then the requirement of SNR amplification becomes even more urgent.        (B) The noises inside the silo.        
(ii) In many cases, it is desirable that the transmission of the signal terminate before the process of acquisition of the signal starts at the receiver. Otherwise, the strong transmitted signal leaks into the acoustic receiver as a strong noise added to the possible weak received reflected signal of interest whose parameters are to be measured. This complicates the processing significantly. If transceivers serve both as transmitters and receivers for beamforming, as in U.S. Ser. No. 10/571,693, it also reduces the number of transceivers usable as receivers, since several transceivers will be still used for transmission; it also makes impossible the use of all transceivers for the simultaneous transmission to create a spatially sharp beam. Thus, for these and other technological reasons the separation of the transmission and the reception in time is essential. This results in the demand that the length of the transmitted pulse [equal to the time period of its transmission multiplied by the speed of sound] be slightly shorter than the distance to the upper surface of the silo contents. This constraint is essential to the silo measurements, while it is absent e.g. from “standard” applications such as RF radar where targets such as airplanes, are far away from the rangefinder.
(iii) Typically, as shown in FIG. 2, the transmitted acoustic signal 110 in silo 100 returns back to receivers 106 not as a single pulse but as a sum [or “train”] of multiple reflections 112, 114. This multi-path phenomena is also known in musical acoustics as reverberation. It presents challenges to discrimination of different reflections [in particular, for point mapping upon the upper surface of the silo contents]. In particular, it would be desirable to have a short-in-time signal, without long tails or ripples, since the tail or ripple of one reflected signal may mix with another reflected signal and cover or “camouflage” the other reflected signal, thus making discrimination of the two reflected signals problematic or impossible. Thus: good separation of the received pulses in-time [after their pre-processing] is required, for time of arrival (TOA) determination and DOA determination and the output pulses [again, after pre-processing] need to be to be as short as possible in time to achieve good resolution and to have tails with very small peak-to-ripple ratios. This challenge is addressed below by an innovative design of the transmitted pulse, and innovative pulse processing.
The term “pre-processing” means the intermediate processing step. In typical prior art general radar or communication systems this includes application of e.g. Matched Filtering (MF) [correlation of the signal pulse with itself, which is an example of pulse compression] or application of other type of filtering which reduces the side-lobes, e.g. mis-matched filtering.
(iv) In addition, the transmitted acoustic pulse has to be band-limited. This follows from several reasons. The first reason is related to the transfer functions of the acoustic antenna (for example, a horn antenna) and of the [electro-mechanical] driver of the acoustic transducer, since the transmission in the frequency domain of high attenuation of the total transfer function is a waste of energy and causes warping of the signal shape. The second reason involves the acoustic noise in silos. Typically the noise is expected to be high at the low frequencies [usually less than 1 kHz] and thus the signal spectrum needs to be separated from these areas.
Note, that the Doppler effect of the signal frequency shift is not an issue for an acoustic rangefinder in a typical silo due to the slow movement of the material surface and the kilo-Hertz range of the carrier frequency. For example, for 1 cm/sec movement and a 5 kHz carrier, and assuming a speed of sound of 340 m/sec, the frequency shift is about 0.01/340*5000=0.14 Hz, which is negligible.
The following notation is used below:
(A) The Z-Transform and Sparsity
The standard notation for the z-transform of a sequence is used herein. For example, the z-transform of the Barker 5 sequence, [1 1 1 −1 1], isb5(z)=1+z−1+z−2−z−3+z−4 The notation b5(z30) means:b5(z30)=1+z−30+z−60−z−90+z−120 This is the z-transform of a sparse sequence of length 121 with only five non-zero values equidistant from one another. The value 30 is the “sparsity” of the new sequence that has been created from the original sequence [1 1 1 −1 1]. The original Barker 5 sequence has a sparsity of 1. Such original sequences of unit sparsity also are referred to herein as “templates” from which sparse sequences are derived. In this example, b5(z) is the template of b5(z30).
(B) Convolution
Several notations are used herein for convolution. All these notations are standard in the signal processing literature.
“” means convolution. For example, if s1=[1, 2] and s2=[1, 2, 3] then c=s1[1, 4, 7, 6]. An alternative notation is are c=conv(s1, s2). In the z-transform domain, the z-transform of c is the product of the z-transforms of s1 and s2.
(C) Matched Filter
MF( . . . ) means a matched filtering operation on a sequence. For example, for a sequence [a b c d e], MF([a b c d e])=[e* d* c* b* a*]; where “*” means conjugation for complex-valued components. For real-valued sequences, MF( . . . ) is equivalent to time-reversal (or index-reversal) of the sequence.
The following references provide background for the prior art of ranging pulse construction and processing:    [BARKER 1953] Barker, R. H: “Group synchronizing of binary digital system” in Jackson, W, (Ed): Communication theory (Butterworths, London, 1953), pp. 273-287.    [BORWEIN 2008] Borwein P., and Mossinghoff M. J., Barker sequences and flat polynomials (with P. Borwein), Number Theory and Polynomials (Bristol, U.K., 2006), J. McKee and C. Smyth, eds., London Math. Soc. Lecture Note Ser. 352, Cambridge Univ. Press, 2008.    [CHEN 2002] Chen, R., and Cantrell B: “Highly bandlimited radar signals”, Proceedings of the 2002 IEEE Radar Conference, Long Beach, Calif., Apr. 22-25, 2002, pp. 220-226.    [LEVANON 2004] Levanon N., and Mozeson E: Radar Signals, Wiley & Sons, New York, 2004, xiv+411 pp.    [LEVANON 2005] Levanon N: “Cross-correlation of long binary signals with longer mismatched filters”, IEE Proc.—Radar, Sonar and Navigation, 152 (6), 372-382, 2005.    [NATHANSON 1999] Nathanson F. E., Reilly J. P., cohere M. N.: “Radar Design Principles Signal Processing and the Environment”, 2nd edition, SciTech Publishing, 1999, 720 pp.
Conventional Construction of a Phase-Coded Ranging Pulse
In the baseband (“BB”) representation, a phase coded ranging pulse is represented as a weighted sum of time-continuous baseband shapes [or functions] Ψ(t/tb) which are equally distanced from each other by “bit” time tb [LEVANON 2004, Section 6]:
                                          u            BB                    ⁡                      (            t            )                          =                              ∑                          m              =              1                        M                    ⁢                                          ⁢                                    s              m                        ·                                          Ψ                BB                            ⁡                              (                                                      t                                          t                      b                                                        -                                      (                                          m                      -                      1                                        )                                                  )                                                                        (                              eq            .                                                  ⁢            1                    ⁢                                          ⁢          A                )            This signal is a continuous function of time t. The digital realization of the signal is sampled with a periodicity of Ts≦tb. The “signal bit length” Lb is the number of samples per bit time, Lb=tb/Ts, rounded to the nearest integer.
The weight [or spreading] sequence, s={sm}, m=1:M, is typically a sequence with good correlation properties. In general, the shape function and the digital sequence may be real or complex valued. The sequence s does not have to be binary; the term “bit” time is thus used for historical reasons, since the first (and still widely used) spreading sequences are the Barker binary codes.
The baseband pulse uBB(t) needs to be up-converted at the transmitter to the carrier frequency fc by using standard techniques. It then needs to be down-converted at the receiver.
For the specific but very common case of a real-valued spreading sequence s and for real-valued shapes, one may write the transmitted pulse u(t), modulated by a carrier frequency fc in a way similar to (eq. 1A):
                              u          ⁡                      (            t            )                          =                              ∑                          m              =              1                        M                    ⁢                                          ⁢                                    s              m                        ·                          Ψ              ⁡                              (                                                      t                                          t                      b                                                        -                                      (                                          m                      -                      1                                        )                                                  )                                                                        (                              eq            .                                                  ⁢            1                    ⁢                                          ⁢          B                )            where the “passband shape” Ψ is ΨBB multiplied by a sinusoid of frequency fc:
                              Ψ          ⁡                      (                          t                              t                b                                      )                          =                                            Ψ              BB                        ⁡                          (                              t                                  t                  b                                            )                                ·                      sin            ⁡                          (                                                2                  ⁢                  π                  ⁢                                                                          ⁢                                      f                    c                                    ⁢                  t                                +                                  ϕ                  0                                            )                                                          (                              eq            .                                                  ⁢            1                    ⁢                                          ⁢          C                )            and φ0 is an arbitrary phase. The phase can be chosen, for example, in such a way that Ψ(t/t0)=0 at the earliest time t for which Ψ(t/t0) is defined [for example if Ψ(t/t0) starts at t=0, one may choose φ0=0].
As described above, the digitally sampled baseband pulse uBB(t) is up-converted to the carrier frequency fc. Alternatively, the passband pulse u(t) is sampled digitally and converted directly to the transmitted analog signal by direct digital-to-analog conversion.
The “kernel” (defined below) illustrated in FIG. 5 below is an example of a sampled passband shape created by multiplying a Kaiser window baseband shape by a sinusoid.
Shape Functions and Bandwidth Considerations for Ranging Pulses
The simplest example of a shape function is the rectangular function, which has support, i.e., is non-zero, only in the interval t=[0,tb] [LEVANON 2004, Section 6, p. 100]:
                                          Ψ            BB                    ⁡                      (                          t                              t                b                                      )                          =                              rect            ⁡                          (                              t                                  t                  b                                            )                                =                      {                                                            1                                                                                            for                      ⁢                                                                                          ⁢                      0                                        ≤                    t                    ≤                                          t                      b                                                                                                                    0                                                                                            for                      ⁢                                                                                          ⁢                      t                                        <                                          0                      ⁢                                                                                          ⁢                      or                      ⁢                                                                                          ⁢                      t                                        >                                          t                      b                                                                                                                              (                              eq            .                                                  ⁢            2                    ⁢                                          ⁢          A                )            
This leads to:
            u      BB        ⁡          (      t      )        =            ∑              m        =        1            M        ⁢                  ⁢                  s        m            ·              rect        ⁡                  (                                    t                              t                b                                      -                          (                              m                -                1                            )                                )                    
The rectangle shape attenuates frequency very poorly. The power spectrum of the pulse behave as a sine function [see NATHANSON 1999, pp. 544, 555]:
      G    ⁡          (      f      )        =                    sin        2            ⁡              (                  π          ⁢                                          ⁢                      ft            b                          )                            (                  π          ⁢                                          ⁢                      ft            b                          )            2      Therefore, smoother functions need to be introduced to satisfy the bandwidth constraints.
One solution that applies to this problem is described in [LEVANON 2004, section 6.8, pp. 145-155], which cites the original article [CHEN 2002] and presents the shape as a Gaussian-windowed sine function: [LEVANON 2004, p. 151, equation 6.25]:
                                          Ψ            BB                    ⁡                      (                          t                              t                b                                      )                          =                  {                                                                                                                                        sin                        ⁡                                                  (                                                      π                            ⁢                                                          t                                                              t                                b                                                                                                              )                                                                    ·                      exp                                        ⁢                                          {                                                                        -                                                      1                                                          2                              ⁢                                                              σ                                2                                                                                                                                    ⁢                                                                              (                                                          t                                                              t                                b                                                                                      )                                                    2                                                                    }                                                                            π                    ⁡                                          (                                              t                                                  t                          b                                                                    )                                                                                                                                        for                    ⁢                                                                                  -                                          2                      ⁢                                                                                          ⁢                                              t                        b                                                                              ≤                  t                  ≤                                      2                    ⁢                                                                                  ⁢                                          t                      b                                                                                                                          0                                                                                  for                    ⁢                                                                                  ⁢                    t                                    <                                                            -                      2                                        ⁢                                                                                  ⁢                                          t                      b                                        ⁢                                                                                  ⁢                    or                    ⁢                                                                                  ⁢                    t                                    >                                      2                    ⁢                                                                                  ⁢                                          t                      b                                                                                                                              (                              eq            .                                                  ⁢            2                    ⁢                                          ⁢          B                )            In the above references, the parameter σ is suggested to be chosen as σ=0.7. The parameter tb is chosen to satisfy the bandwidth constraints [the spectrum vs. non-dimensional coordinate f−tb is shown in FIG. 6.42 on p. 153 and 6.45 on p. 154. of [LEVANON 2004]].
The above shapes all have time-support, i.e. they are non-zero, upon a [small, typically 1 or 2 or 4] integral number of tb intervals. This means that for the sampled signals, the signal bit length is equal to the length of the sampled shape function sequence for the rectangular shape function of equation (2A) and to one-quarter of the length of the sampled gaussian-windowed sine shape function of equation 2B.
Some Known Sequences with Good Correlation Properties
A typical ranging pulse is based upon a sequence with good correlation properties [see LEVANON 2004, Chapter 6, Sections 6.1, 6.2, 6.3, 6.4, 6.5, 6.7]. For example, the most celebrated ones are the Barker sequences introduced in 1953 by Barker [BARKER 1953]:
b3=[1,1,−1]
b4A=[1,1,−1,1]
b4B=[1,1,1,−1]
b5=[1,1,1,−1,1]
b7=[1,1,1,−1,−1,1,−1]
b11=[1,1,1,−1,−1,−1,1,−1,−1,1,−1]
b13=[1,1,1,1,1,−1,−1,1,1,−1,1,−1,1]
Each of these sequences has a relatively flat spectrum and also the exceptional property that the peak-to-maximal-ripple-ratio of its auto-correlation is equal to the length of the sequence. This ratio, when calculated in dB, is known in the radar literature as PSL [Peak Sidelobe Level]. For example for b5, this ratio is 5 to 1 [or PSL=10 log10(⅕2)≈−14 dB]. The PSLs of the Barker codes are shown in [NATHANSON 1999, Table 12.1, p. 538].
All known Barker codes are listed above. To enlarge the pulse energy the longer pulses, that have more sequence elements, are used. Still, it is assumed that the autocorrelation properties of such sequences are good. One may find discussion of such codes and some examples e.g. in [LEVANON 2004, Chapter 6, Table 6.3 pp. 108-109], where the sidelobe level is shown, while the peak level corresponds to the length of the sequence; see also [NATHANSON 1999, pp. 537-541, especially the Table 12.2 on pp. 540], where also the number of different possible sequences having the same sidelobe level is presented].
Modern mathematical review of generalizations of Barker sequences may be found in [BORWEIN 2008]. In the mathematical literature, the Littlewood polynomials [polynomials with coefficients ±˜1] are discussed. As an example, the following two sequences, representing the Littlewood polynomials with good autocorrelation properties, are listed in [BORWEIN 2008] and are given as:
seq20=[1,1,1,1,1,−1,1,−1,−1,−1,1,−1,1,1,−1,−1,−1,1,1,−1]
seq25=[1,1,1, −1, −1, −1,1,1,1,1,1,1,1, −1,1, −1,1, −1, −1,1, −1,1,1, −1]
Nested Sequences
A well-known way to construct longer sequences is to use nested codes [see LEVANON 2004, section 6.1.2, pp. 107-109, and LEVANON 2005]. This method is also known as code concatenation (or combination); if two Barker codes are used, the code is also named Barker-squared, see [NATHANSON 1999, pp. 541-542]. For example, instead of 1's and −1's in the binary code, such as a Barker code or a Littlewood polynomial, one may insert another code. To illustrate this approach let us use Barker b5 as a template, and an arbitrary Barker sequence “b” of length “L”; then one may write a sequence of length L·5:nested_sequence—Lx5=[b,b,b,−b,b]
Typically, a Barker code is nested inside of another Barker code. As an example, one may use Barker 5 as the “b” sequence, b=b5, and obtain the 5×5 nested Barker code of length 25:nested_sequence—5×5=[b5,b5,b5,−b5,b5]
The nested code obtained by nesting Barker b13 into itself, 13×13 is mentioned in the literature; see for example [LEVANON 2005]. An example of a very long code using four combined Barker codes: 5×13×13×13 (with total length 10,985) is mentioned in [NATHANSON 1999, p. 542].
Note that nesting does not improve the peak-to-maximum-ripple ratios in the autocorrelation of the nested code [LEVANON 2004]. For example the auto-correlation of b5 nested into b13 (or vice versa) consists of values {0, 1, 5, 13, 65}, hence the peak to max ripple ratio is equal to 5[65/13=5].
Thus the nested code enlarges the pulse energy but does not improve the correlation properties [the PSL value].
Pulse Processing
Pulse compression is achieved by convolution of the received signal with the matched filter of the transmitted signal. This shrinks the long M-element based signal to one energetic main-lobe [which represents the auto-correlation of the shape-function and accumulates the energy of the long pulse] surrounded by side-lobes. For example, if the rectangular shape is used (see eq. 2A) together with the sequence Barker 13, b13, than the well known saw-shape appears as the result of this convolution: it contains one triangular main-lobe and six identical triangular side-lobes to the left and right of the main-lobe. The relation of the peak amplitude of the main-lobe to the peak of each side-lobe is 13/1 as it has to be when the Barker code of length 13 is used [see NATHANSON 1999, FIG. 12.2 on p 536. This example of pulse compression is shown in FIG. 3].
Note also that the main-lobe typically becomes wider than the original shape, as a side-effect of the matched filtering. For example, in the above example the triangle of the main-lobe occupies the time interval [−tb,tb], whereas the original rectangular shape-function occupies the time interval [0,tb].
To suppress the side-lobes, mis-matched filtering is introduced [see LEVANON 2004, Section 6.6 pp. 140-142]. For this purpose an especially pre-calculated sequence, “q” of length K is used. This digital sequence is prepared such that the convolution of the original spreading sequence s with the mismatched sequence q,g=sq is close to a “delta function”, i.e., it has one large value (“the peak”) and all other values are small. For example, [LEVANON 2005] demonstrates that a filter that is three times longer than the original Barker 13 [i.e. a mis-matched filter of length 3*13=39] leads to a PSL ratio better than −40 dB (specifically, −43.241 dB). [NATHANSON 1999, Section 12.4 pp. 555-559] mentions that the PSL level can be reduced as low as one desires. For example, a plot for the PSL level vs. mis-matched filter length is shown in [NATHANSON 1999, FIG. 12.10, p. 557] and demonstrates PSL about −55 dB for Barker 13 and a mis-matched filter of length about 63.
Using a mis-matched filter leads to some SNR loss, which is relatively small for Barker 13 [tenths of a dB, for example just 0.2 dB according to [LEVANON 2005]]. The SNR loss may be larger for other spreading sequences [NATHANSON 1999, p. 557].
The application of very long mis-matched filters to nested binary codes is discussed in [LEVANON 2005]. For example, a mismatched filter of length 507 is applied to a 13×13 Barker nested code. The length is chosen to be three times the total length of the nested code: 507=3×3×13. The PSL attained is about −40 dB.
For the case of rectangular shape functions and nested binary codes, a special mechanism based on several digital signal processors for mismatched filtering is discussed in [NATHANSON 1999, pp. 571-573].